NMR data processing may involve averaging over a finite window of measured samples, centered over the expected peak of each spin echo, to estimate the amplitudes of individual spin echoes. The result of this averaging procedure is dependent on the shape of the echo. For example, if the echo amplitude and phase are perfectly constant over the duration of the averaging window the calculated average will provide an accurate estimate of the peak echo amplitude. If however, the amplitude and/or phase of the echo changes significantly over the averaging window, then the calculated average will significantly underestimate the peak echo amplitude. This underestimation effect can be expected when using a rectangular averaging window or a time-domain-weighted averaging function such as a Hanning window.
The time domain shape of a spin echo may be controlled by the distribution of static magnetic field gradients across the spin-density volume of the sample under investigation. If the static magnetic field exhibits smaller gradients across the sample of interest, the echo shape tends to be generally broader and flatter across the fixed length time domain averaging window. If the static magnetic field exhibits larger gradients across the same sample of interest, the echo shape tends to be generally narrower with more rapid dephasing on either side of the echo peak. In addition some NMR measurement effects can cause the echo peak to shift in time, and this can also affect the time domain average as calculated over a fixed echo averaging window.
In many practical applications the magnetic field gradients across a sample of interest are unknown and outside the control of the NMR measurement device or the user. For example, naturally-occurring soil and rock samples often have significant magnetic mineral content, and this content can induce significant gradients in the static magnetic field of the measurement device across and within the volume of the sample under investigation. For applications involving detection of water, oil or other hydrogen-rich fluids these internal sample-induced gradients cause the NMR resonance frequency (the Larmor frequency) of the fluids to vary significantly within the sample volume, and this causes a sharpening of the echo shape. An example of this effect is shown in FIG. 1, which compares the measured NMR spin echo shapes for bulk water and seven water-saturated soil samples with different bulk magnetic susceptibilities. These samples were all measured under identical conditions in the same 275 kHz laboratory spectrometer. The samples with relatively high bulk magnetic susceptibility (Labeled “Rifle”, “Hanford” and “Alamosa”) exhibit significantly sharper and narrower echo shapes than the other samples that have lower bulk magnetic susceptibility. This effect can occur for laboratory NMR spectrometers which are designed to have low gradients, for NMR logging tools that often have high gradients, and also for surface NMR instruments that perform NMR measurements in the Earth's magnetic field. These internal gradient effects can also affect medical magnetic resonance imaging measurements, in particular in and around volumes of tissue or objects that induce magnetic susceptibility contrasts.
For some NMR measurement devices, the NMR amplitude response of the device may be calibrated under some known, nominal measurement condition. For example, with a laboratory NMR spectrometer, one can calibrate the NMR amplitude response for a known sample of bulk water or some other fluid or solid. Or for example, one can calibrate the NMR amplitude response of an NMR logging tool by immersing the tool in a volume of water or other fluid or solid that is known to incorporate the entire NMR sensitive volume of the tool. In such calibration schemes, the peak echo amplitude may be estimated for each echo by averaging the recorded data over a finite window of samples centered on the location of the expect peak of each echo. Hence, these commonly-performed calibration methods do not account for the distortions of the spin echo shape that are caused by unknown internally-induced or externally-applied magnetic field gradients in actual sample measurement conditions. And thus, these actual sample measurement conditions, coupled with windowing and time-averaging methods, often result in under-estimation of actual measured echo amplitudes. This in turn leads to underestimation of the underlying spin density, and in particular this can lead to underestimation of sample fluid content.